Bourgin-Yang versions of the Borsuk-Ulam theorem for $p$-toral groups

TL;DR

This paper extends the Bourgin-Yang theorem to $p$-toral groups, providing dimension estimates for zero sets of equivariant maps between spheres of orthogonal representations, and shows infinite-dimensional zero sets in certain cases.

Contribution

It generalizes the classical Bourgin-Yang theorem to $p$-toral groups and establishes new dimension bounds for zero sets of equivariant maps.

Findings

Provides dimension estimates for zero sets under $p$-toral group actions.

Extends classical Bourgin-Yang theorem to new classes of groups.

Shows infinite-dimensional zero sets for certain infinite-dimensional representations.

Abstract

Let $V$ and $W$ be orthogonal representations of $G$ with $V^G= W^G=\{0\}$. Let $S(V )$ be the sphere of $V$ and $f : S(V ) \to W$ be a $G$-equivariant mapping. We give an estimate for the dimension of the set $Z_f=f^{-1}\{0\}$ in terms of $ \dim V$ and $\dim W$, if $G$ is the torus $\mathbb T^k$, or the $p$-torus $\mathbb Z_p^k$. This extends the classical Bourgin-Yang theorem onto this class of groups. Finally, we show that for any $p$-toral group $G$ and a $G$-map $f:S(V) \to W$, with $\dim V=\infty$ and $\dim W<\infty$, we have $\dim Z_f= \infty$.